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G. Felder

Publications and source records attributed to G. Felder.

At least 19 recordsLinked to original sources
arXiv2023

Hypergeometric integrals, hook formulas and Whittaker vectors

We determine the coefficient of proportionality between two multidimensional hypergeometric integrals. One of them is a solution of the dynamical difference equations associated with a Young diagram and the other is the vertex integral associated with the Young diagram. The coefficient of proportionality is the inverse of the product of weighted hooks of the Young diagram. It turns out that this problem is closely related to the question of describing the action of the center of the universal enveloping algebra of $\mathfrak{gl}_n$ on the space of Whittaker vectors in the tensor product of dual Verma modules with fundamental modules, for which we give an explicit basis of simultaneous eigenvectors.

math-ph↗
arXiv2011

Zeroes of Wronskians of Hermite polynomials and Young diagrams

For a certain class of partitions, a simple qualitative relation is observed between the shape of the Young diagram and the pattern of zeroes of the Wronskian of the corresponding Hermite polynomials. In the case of two-term Wronskian $W(H_n, H_{n+k})$ we give an explicit formula for the asymptotic shape of the zero set as $n \rightarrow \infty$. Some empirical asymptotic formulas are given for the zero sets of three and four-term Wronskians.

math-ph↗
arXiv2007

Equivariant Lefschetz number of differential operators

Let $G$ be a compact Lie group acting on a compact complex manifold $M$. We prove a trace density formula for the $G$-Lefschetz number of a differential operator on $M$. We generalize Engeli and Felder's recent results to orbifolds.

math.QA↗
arXiv2005

Poincare-Birkhoff-Witt expansions of the canonical elliptic differential form

We study the canonical U(\n)-valued elliptic differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of elliptic KZ differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebras A_r, B_r, C_r, D_r in a conveniently chosen Poincare-Birkhoff-Witt basis. As an application we give a new formula for eigenfunctions of Hamiltonians of the Calogero-Moser model.

math.RT↗
arXiv2002

Multiplication Formulas for the Elliptic Gamma Function

The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma function, whose degenerations are the Gauss-Askey multiplication formula for the Euler and trigonometric gamma functions.

math.QA↗
arXiv2000

The geometry of WZW branes

The structures in target space geometry that correspond to conformally invariant boundary conditions in WZW theories are determined both by studying the scattering of closed string states and by investigating the algebra of open string vertex operators. In the limit of large level, we find branes whose world volume is a regular conjugacy class or, in the case of symmetry breaking boundary conditions, a `twined' version thereof. In particular, in this limit one recovers the commutative algebra of functions over the brane world volume, and open strings connecting different branes disappear. At finite level, the branes get smeared out, yet their approximate localization at (twined) conjugacy classes can be detected unambiguously. As a by-product, it is demonstrated how the pentagon identity and tetrahedral symmetry imply that in any rational conformal field theory the structure constants of the algebra of boundary operators coincide with specific entries of fusing matrices.

hep-th↗
arXiv2000

Differential Equations Compatible with KZ Equations

We define a system of "dynamical" differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra $\mathbf{g}$. These are equations on a function of $n$ complex variables $z_i$ taking values in the tensor product of $n$ finite dimensional $\mathbf{g}$-modules. The KZ equations depend on the "dual" variable in the Cartan subalgebra of $\mathbf{g}$. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions.

math.QA↗
arXiv2000

Solutions of the KZB equations in genus \geq 1

We introduce a flat version of the KZB connection. This connection is defined on the complement of the locus of Weierstrass points on the moduli space of genus $g$ complex curves with marked points. We then give integral formulas for flat sections of this connection, using the parametrization of conformal blocks and of the KZB connection of our earlier paper math/9807145.

math.QA↗
arXiv1999

Conformal boundary conditions and three-dimensional topological field theory

We present a general construction of all correlation functions of a two-dimensional rational conformal field theory, for an arbitrary number of bulk and boundary fields and arbitrary topologies. The correlators are expressed in terms of Wilson graphs in a certain three-manifold, the connecting manifold. The amplitudes constructed this way can be shown to be modular invariant and to obey the correct factorization rules.

hep-th↗
arXiv1998

Commuting differential and difference operators associated to complex curves, II

We construct a commuting family of difference-evaluation operators, deforming the commuting family introduced in our earlier paper (math/9807145). We interpret them as the action of the center of quantum algebras in the space of intertwiners for a ``regular'' subalgebra. These algebras are the quantum groups associated with sl_2 and algebraic curves, introduced by V. Rubtsov and the first author (q-alg/9608005). In the case of rational curves, our operators coincide those provided by the Yangian action on the hypergeometric spaces of Tarasov and Varchenko (q-alg/9604011).

math.QA↗
arXiv1998

Commuting differential and difference operators associated to complex curves, I

We introduce twists by Cartan elements of conformal blocks on a curve X, corresponding to a Lie algebra g. We show that these twists define holomorphic functions, with theta-like behaviour, on a product of copies of its Jacobian J(X)^r. We then parametrise the conformal blocks by their twisted correlation functions of nilpotent currents, in the spirit of Feigin-Stoyanovsky. We compute the action of the Sugawara tensor in terms of these correlation functions. The result is a family of operators acting on forms on the product J(X)^r \times \prod_{i} S^{n_i}X of the Jacobian of X with its symmetric powers. These operators act by differentiation in the Jacobian directions, and evaluations and residues in the variables of the symmetric powers. They serve to express the KZB connection, which we view as a connection on the space of curves with marked a-cycles. These operators also make sense when the level is critical. In that case, they commute with each other. We explain their connection with the Beilinson-Drinfeld operators. In a sequel to this paper, we will do a similar computation in the q-deformed case, replacing the inclusion of the enveloping algebras of the Lie algebra of regular functions on X in that of functions on the formal disc by some inclusion of quasi-Hopf algebras, which were introduced in work of one of us and V. Rubtsov. The outcome is a commuting family of difference-evaluation operators, which may be viewed in the rational case as the Bethe ansatz formulation of the qKZ operators.

math.QA↗
arXiv1997

Coinvariants for Yangian doubles and quantum KZ equations

We present a quantum version of the construction of the KZ system of equations as a flat connection on the spaces of coinvariants of representations of tensor products of Kac-Moody algebras. We consider here representations of a tensor product of Yangian doubles and compute the coinvariants of a deformation of the subalgebra generated by the regular functions of a rational curve with marked points. We observe that Drinfeld's quantum Casimir element can be viewed as a deformation of the zero-mode of the Sugawara tensor in the Yangian double. These ingredients serve to define a compatible system of difference equations, which we identify with the quantum KZ equations introduced by I. Frenkel and N. Reshetikhin.

q-alg↗
arXiv1997

Elliptic quantum groups $E_{τ,η}(sl_2)$ and quasi-Hopf algebras

We construct an algebra morphism from the elliptic quantum group $E_{τ,η}(sl_2)$ to a certain elliptic version of the ``quantum groups in higher genus'' studied by V. Rubtsov and the first author. This provides an embedding of $E_{τ,η}(sl_2)$ in an algebra ``with central extension''. In particular we construct $L^{\pm}$-operators obeying a dynamical version of the Reshetikhin--Semenov-Tian-Shansky relations. To do that, we construct the factorization of a certain twist of the latter algebra, that automatically satisfies the ``twisted cocycle condition'' of O. Babelon, D. Bernard and E. Billey, and therefore provides a solution of the dynamical Yang-Baxter equation.

q-alg↗
arXiv1997

A construction of Hopf algebra cocycles for the double Yangian $DY(SL_{2})$

We construct a Hopf algebra cocycle in the Yangian double $DY(SL_{2})$, conjugating Drinfeld's coproduct to the usual one. To do that, we factorize the twist between two ``opposite'' versions of Drinfeld's coproduct, introduced in earlier work by V. Rubtsov and the first author, using the decomposition of the algebra in its negative and non-negative modes subalgebras.

q-alg↗
arXiv1992

Gravity in Non-Commutative Geometry

We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.

hep-th↗
arXiv1992

Grand Unification in Non-Commutative Geometry

The formalism of non-commutative geometry of A. Connes is used to construct models in particle physics. The physical space-time is taken to be a product of a continuous four-manifold by a discrete set of points. The treatment of Connes is modified in such a way that the basic algebra is defined over the space of matrices, and the breaking mechanism is planted in the Dirac operator. This mechanism is then applied to three examples. In the first example the discrete space consists of two points, and the two algebras are taken respectively to be those of $2\times 2$ and $1\times 1$ matrices. With the Dirac operator containing the vacuum breaking $SU(2)\times U(1)$ to $U(1)$, the model is shown to correspond to the standard model. In the second example the discrete space has three points, two of the algebras are identical and consist of $5\times 5$ complex matrices, and the third algebra consists of functions. With an appropriate Dirac operator this model is almost identical to the minimal $SU(5)$ model of Georgi and Glashow. The third and final example is the left-right symmetric model $SU(2)_L\times SU(2)_R\times U(1)_{B-L}.$

hep-ph↗